Problem 30 - Algebra 2 and Trigonometry

Question

In \(26-37,\) find each product. $$ (7+i)(2+3 i) $$

Step-by-Step Solution

Answer

The product is \(11 + 23i\).

Step 1: Apply the Distributive Property

To find the product \((7+i)(2+3i)\), we'll use the distributive property, also known as the FOIL method for binomials, which stands for First, Outer, Inner, Last. Multiply each term in the first binomial by each term in the second.

Step 2: Multiply the First Terms

Multiply the first terms from each binomial: \(7 \cdot 2 = 14\). This is the First product.

Step 3: Multiply the Outer Terms

Multiply the outer terms from each binomial: \(7 \cdot 3i = 21i\). This is the Outer product.

Step 4: Multiply the Inner Terms

Multiply the inner terms from each binomial: \(i \cdot 2 = 2i\). This is the Inner product.

Step 5: Multiply the Last Terms

Multiply the last terms from each binomial: \(i \cdot 3i = 3i^2\).Since \(i^2 = -1\), we have \(3i^2 = 3(-1) = -3\). This is the Last product.

Step 6: Combine the Products

Now, combine all the products: \(14 + 21i + 2i - 3\).

Step 7: Simplify the Expression

Combine the like terms: \(14 - 3 = 11\) for the real components, and \(21i + 2i = 23i\) for the imaginary components.Therefore, the expression simplifies to \(11 + 23i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics that allows us to simplify expressions and solve equations. It can be expressed as:

• For any numbers or expressions, \( a(b + c) = ab + ac \).
• This means you multiply the term outside the parentheses by each term within.

This property ensures that when dealing with expressions like products of binomials, each term in the first binomial is multiplied by every term in the second.
In the exercise given, we utilize the distributive property to expand the expression \((7+i)(2+3i)\).
This thorough approach enables us to handle more complex expressions by transforming them into simpler forms suitable for further simplification.

FOIL Method

The FOIL method is a specific instance of the distributive property used when multiplying two binomials. FOIL stands for:

• First: multiply the first terms of each binomial.
• Outer: multiply the outer terms.
• Inner: multiply the inner terms.
• Last: multiply the last terms.

This method ensures that no pair of terms is neglected.
In our example, when we multiply \((7+i)(2+3i)\), we follow the FOIL method step by step to ensure accuracy. We begin with multiplying first terms, then outer, inner, and finally last terms, which is essential in deriving the correct result.
This structured approach simplifies the process, especially when dealing with complex expressions.

Imaginary Unit

The imaginary unit, denoted as \(i\), is used to extend the real number system. It serves as the foundation for complex numbers.
The defining property of the imaginary unit is:

• \(i^2 = -1\).

This unique property allows exploration beyond the limitations of real numbers, providing solutions to equations that have no real roots.
In our calculation, when multiplying terms involving \(i\), such as \(i \times 3i\), we encounter \(i^2\).
By substituting \(i^2 = -1\), we convert complex products into simpler real components, situated firmly within the realm of real numbers.

Binomials

Binomials are algebraic expressions composed of two distinct terms linked by addition or subtraction.

• Examples of binomials include \(a + b\) or \(x - y\).
• These expressions play a vital role in algebra, particularly when learning multiplication techniques such as the distributive property and FOIL method.

When multiplying binomials, as shown in this exercise with \((7+i)(2+3i)\), it's crucial to apply these techniques methodically.
Essentially, each term in the first binomial must be multiplied by every term in the second to ensure a correct expansion.
This practice prepares students for tackling more advanced algebraic problems involving complex numbers and beyond.